Generalized fuzzy c-shells clustering and detection of circular and elliptical boundaries

The Fuzzy c-Shells (FCS) algorithm and its adaptive generalization, called the Adaptive Fuzzy c-Shells (AFCS) algorithm, are considered for detection of curved boundaries, specifically circular and elliptical. The FCS algorithms utilize hyper-spherical-shells as cluster prototypes. Thus in two dimensions, the prototypes are circles. The AFCS algorithms consider hyper-ellipsoidal-shells as prototypes, hence the ability to characterize elliptical boundaries. The generalization is achieved by allowing the distances to be measured through a norm inducing matrix that is symmetric, positive definite. Each cluster is allowed to have a different matrix, which is made a variable of optimization. The ability of the algorithms to detect circular and elliptical boundaries in two-dimensional data is illustrated through several examples.